Joint PDF probability problem

iocal said:

Hi guys,

I am looking to confirm whether I have got this one right.
Let x and y have the joint pdf \(\displaystyle \ f(x,y)=6(1-x-y),\ x+y < 1,\ 0< x,0<y,\ \mathrm zero\ elsewhere\)
Compute \(\displaystyle \ P(2x+3y<1)\)

What I thought is right was to compute the probability with the following limits of integration:

\(\displaystyle \displaystyle \int_0^{\frac{1-3y}{2}} \left[ \int_0^{\frac {1}{2}} 6(1-x-y)\ \mathrm dy\right]\ dx\)

Have I got the limits right? If not, could you please tell where I went wrong?
Thanks.

Click to expand...

DrPhil said:

I added a set of [ ] to make what you have written completely clear. The first (inner) integration has to be the one with a variable in the limits. If you integrate first with respect to \(\displaystyle y\), then the limits are from \(\displaystyle y=0\) to \(\displaystyle y=(1-2x)/3\). That way, after the integration you will have a function of \(\displaystyle x\) only.

\(\displaystyle \displaystyle \int_0^{1/2} \left[ \int_0^{(1-2x)/3} 6(1-x-y)\ \mathrm dy\right]\ \mathrm dx\)

Click to expand...